Humboldt-Universität, Berlin, 25.5.2005
Dieter Bestle and Akin Keskin
Chair of Engineering Mechanics and Vehicle DynamicsBrandenburg University of Technology Cottbus
Some Applications of Multicriterion Strategies in Mechanical Engineering
Humboldt-Universität, Berlin, 25.5.2005
Classification of Optimization Problems
optimization
topologyoptimization
shapeoptimization
sizing
scalaroptimization
multicriterionoptimization
scalarization
hierarchization
unconstrained constrained
)(min pp
fh
RI∈)(min p
pf
P∈
ouhRIP ppp0ph0pgp ≤≤≤=∈= ,)(,)(
Humboldt-Universität, Berlin, 25.5.2005
What is Optimization Good For?
min
l
uP0 -∈
Ω Ω Ω
pmaximum stable speed range
0
0
0: min . . max Re( ) 0
: min . . max Re( ) 0l
lj
j
uj
j
s t
s t
λ
λ
≤ Ω ≤ Ω
Ω < Ω ≤ Ω
Ω = Ω ≤
Ω = Ω >
2 3 4, ,min max ( ) , ( ) , ( )
x tx t x t x t
γ ε∆minimum motion of inner balls
( )min Tρ ϕ
ˆ. . ( )s t s T s=
minimum time
• Designing Birthday Presents
Humboldt-Universität, Berlin, 25.5.2005
What is Optimization Good For?
• Designing Birthday Presents
• Solving Problems of Existing Machines
ideal contact geometry
( )
( )
max
min
r
z
ρ ψ
ρ ψ
δ
δ
0. . tans t ϕ µ≤
ideal ring shape
, ,min T
Sw
γγ +
rr r
0
0
00. . ,/r rk k
r kkrk
sS S
Gt SS
γµ−
≤=
rkS
D
H
S
S
kr
ϕ
( )ρ ψ
rδ
zδ
N
T
Humboldt-Universität, Berlin, 25.5.2005
What is Optimization Good For?
• Designing Birthday Presents
• Solving Problems of Existing Machines
• System Identification
• actuator behavior
(hydraulic, pneumatic, electro-mechanical)
• dynamic behavior of passive components
• system parameters
• control behavior
Humboldt-Universität, Berlin, 25.5.2005
What is Optimization Good For?
• Designing Birthday Presents
• Solving Problems of Existing Machines
• System Identification
• Virtual Prototyping
• hardware-in-the-loop optimization
• mechatronics (control design)
• multibody systems (vibrations)
• multi-disciplinary optimization
Humboldt-Universität, Berlin, 25.5.2005 ouhRIP ppp0ph0pgp ≤≤≤=∈= ,)(,)(
scalaroptimization
vectoroptimization
min f(p)p∈P
min p∈P
f1 (p)...
fn (p)
P
f1
f2
f(P)
1
2
multiple optimal compromises
f(p)
Why Multi-criterion Optimization?
a technical point of view
p1
f
single optimal solution
Humboldt-Universität, Berlin, 25.5.2005
scalaroptimization algorithm
Reduction Principles for Vector Optimization
vector optimization problem
scalar optimization problem
…
obje
ctive
1
obje
ctive
n
constr
ain
t1
…
constr
ain
tm• scalarization
(weighted obj., distance method,
goal attainment)
• hierarchization(hierarchical opt.,
compromise method)
• combination(goal programming)
Humboldt-Universität, Berlin, 25.5.2005
Example 1: Horizontal Platform Insulation
physical modeling
Linearization
Equivalent Mechanical Model (Q=0)
Guxy ∆−∆=∆
problem: increase of horizontal damping
( )
( )
( )212
2101
2
2
2221
1
111
01
,
ppqQV
ppqyAV
VV
pnpV
V
pnp
AppF
L
L
L
−−−=
−+−=
−=−=
−=
ɺ
ɺɺ
ɺɺɺɺ
( ) dtQqyAyqApqp
ApF
LLL ∫+∆+∆+∆+−=∆
∆=∆
αβααββα ɺɺ0011
01
( )2 1 21 2
2 2
k k kF F y k k y
d d∆ + ∆ = ∆ + + ∆ɺ ɺ
Humboldt-Universität, Berlin, 25.5.2005
multi-measurement identification
experimentalsetup
Simulink model
2
0
1 , ,
,
T
i
F Fsim i meas i
where dtT F
meas i
ϕ
−= ∫
0,: ≥=∑ i
i
ii wwu ϕobjectives
1,:
/1
0 ≥
−= ∑ ru
rr
i
ii ϕϕ
optimization(Matlab)
measF simF
Example 1: Horizontal Platform Insulation
Humboldt-Universität, Berlin, 25.5.2005
platform insulation
,ˆ
ˆmin
℘∈ f
H
p,
ˆ
ˆmin
℘∈ T
H
p
℘∈ T
f
ˆ
ˆminp
passive
active
xDQ ɺ∆=stream flow control
60 ≤≤ BVs
bar6bar4 0 ≤≤ p
[ ]TBVsp0=p
0)Re( <iλ
0.0100.012- ≤≤ D
bar6bar4 0 ≤≤ p
[ ]TDp0=p
0)Re( <iλ
bi-criterion problems
1 1.5 2 2.5 3 3.5 40
1
2
3
4
5
0 1 2 3 50.14
0.16
0.18
0.2
0.22
0.24
0.28
1 1.5 2 2.5 3 3.5 40.14
0.16
0.18
0.2
0.22
0.24
0.28
3.26 3.27 3.280.201
0.202
0.203
passive
passive
passive
active
active
active
( )( )ii
T
iHiHf
iHH
λ
ωωπω
ω
ω
ω
Remax
1ˆ
)(max)ˆ(:2/ˆˆ
)(maxˆ
−=
==
=
objectives to be minimized
Example 1: Horizontal Platform Insulation
Humboldt-Universität, Berlin, 25.5.2005
f1
f2
f(P)
*jf
*if
* *: i j = F f f
Fββββ
nFβ t+
n
single EP-solution
)(pfnFβ =+ t
ttP
min,∈p
. .s t
.const=β
)(pfnFβ =+ t
ttP
min,, βp∈
. .s t
1,0 =∑≥ ii ββ
knee search
Normal Boundary Intersection Approach
related to I. Das and J.E. Dennis
recursive knee search
Humboldt-Universität, Berlin, 25.5.2005
2) lower and upper recursion limits
Recursive Knee Search
individual minima
• initial design
initial design
knee search
global optimality
localoptimality
stoppingcriteria
trade-off curve
2/λλ =
fulfilled
fulfilled
recursionloop
• stopping criteria
• bounds on knee search
( ) ( )(0)i j i1 1,1randλ ε∗ ∗ ∗= + + − − p p p p
( )1
0,1 , 0, min 1, 1λ ελ
∈ ∈ −
. . 0.2, 0.5e g ε λ= =
| | , . . 0.1t t e g t≤ =1)
1)
2) min max[ , ], . . [0.4,0.6]e gβ β β β∈ ∈
,]1,1[−∈t
• local optimality
1) solution feasible2) local EP-optimality3) limits on No. of non-successful restarts
Matlab applet
Humboldt-Universität, Berlin, 25.5.2005
VIT-project (Virtual Turbomachinery, LuFo III)
Improvement of current Rolls-Royce compressor design process
by tool integration and optimization
step 1: tool by tool analysis of current compressor design process
step 2: automation and integration of single analysis tools into an optimization environment
step 3: partial automation of multidisciplinary design process
Tool8
Tool7
Tool5
Tool1
Tool2
Tool3
Tool4Tool6
iSight
the future: automation of whole multidisciplinary design process
Example 2: Compressor Design Process
Humboldt-Universität, Berlin, 25.5.2005
Aerodynamics (Keskin)
cw
u
MeanlinePrediction
Throughflow 2D Blading 3D Blading
Design 2D-Thermal
Design and Stress (Otto)
High Cycle and Low CycleFatigue
Humboldt-Universität, Berlin, 25.5.2005
MeanlinePrediction
annulus parameterization
0 5 10 15 20 25 304
6
8
10
12Annulus Line
• classical
point-wise definition
• Reduced parameter
definition
(Bézier-splines
with 4 control points)
0 10 20 305
6
7
8
9Mid-Height Line
0 10 20 300
2
4
6Thickness Line
2 2 3 3 2 2 3 3, , , , , , ,
T
x y x y x y x ydesign parameters b b b b t t t t = p
Matlab applet
Humboldt-Universität, Berlin, 25.5.2005
polyc,max η p
6.0 max ≤iiΨ
1.1max ', ≤RiI
iM
8.0max , ≤SiI
iM
55.0max ≤Ri
iDF
55.0max ≤Si
iDF
58.0max ≥Ri
iDH
58.0max ≥Si
iDH 92.0 max , ≤ih
iC
27.0, ≤S
NE sM
s,N, i …1∈
SM maxp
25SM %≥
MeanlinePrediction
criteria
constraints
Humboldt-Universität, Berlin, 25.5.2005
87.5 88 88.5 89 89.5 90 90.5
Pareto Opitimal Solutions for Meanline Annulus Modification
Datum, Human Designer
iSight, MIGA, 4CP
Pareto Optimal Design
Results for Bézier-splines with 4 control points
E3E
human design
MeanlinePrediction
Polytropic Efficiency, ηηηηpoly
[%]
Su
rge
Ma
rgin
, S
M [
%]
Humboldt-Universität, Berlin, 25.5.2005
0 10 20 305
6
7
8
9Mid-Height Line
0 10 20 300
2
4
6Thickness Line
0 5 10 15 20 25 304
6
8
10
12Annulus Line
4 control points allow
1 turning point, only !E3E has two
turning points
explanation for sub-optimal solution 1. parameterization too restrictive
5 control points
MeanlinePrediction
Humboldt-Universität, Berlin, 25.5.2005
polyc,max η p
6.0 max ≤iiΨ
1.1max ', ≤RiI
iM
8.0max , ≤SiI
iM
55.0max ≤Ri
iDF
55.0max ≤Si
iDF
58.0max ≥Ri
iDH
58.0max ≥Si
iDH 92.0 max , ≤ih
iC
27.0, ≤S
NE sM
s,N, i …1∈
SM maxp
25SM %≥
criteria
constraints
explanation for sub-optimal solution 2. constraints too restrictive
MeanlinePrediction
0.93
0.285
Humboldt-Universität, Berlin, 25.5.2005
Pareto Opitimal Solutions for Meanline Annulus Modification
Datum, Human Designer
iSight, MIGA, 4CP
iSight, MIGA, 5CP
Pareto Optimal Design
Matlab applet
MeanlinePrediction
E3E
human designPolytropic Efficiency, ηηηηpoly
[%]
Su
rge
Ma
rgin
, S
M [
%]
Humboldt-Universität, Berlin, 25.5.2005
E3E
max. efficiency
E3E
max. surge margin
MeanlinePrediction
Humboldt-Universität, Berlin, 25.5.2005
Pareto Opitimal Solutions for Meanline Annulus Modification
Datum, Human Designer
iSight, MIGA, 5CP
iSight, NLPQL, 5CP
Pareto Optimal Design
MeanlinePrediction
E3E
human designPolytropic Efficiency, ηηηηpoly
[%]
Su
rge
Ma
rgin
, S
M [
%]
Humboldt-Universität, Berlin, 25.5.2005
Pareto Opitimal Solutions for Meanline Annulus Modification
Datum, Human Designer
iSight, MIGA, 5CP
iSight, NLPQL, 5CP
Pareto Optimal Design
SM increase 17.3%
ηp increase 0.17%
MeanlinePrediction
E3E
human design Polytropic Efficiency, ηηηηpoly
[%]
Su
rge
Ma
rgin
, S
M [
%]
Humboldt-Universität, Berlin, 25.5.2005
pressure ratio
1 2 3 4 5 6 7 8 91
1.2
1.4
1.6
1.8Parametric Stage Pressure Ratio Distribution
1 2 3 4 5 6 7 8 91
1.2
1.4
1.6
1.8Applied Stage Pressure Ratio Distribution
2 2 3 3 4 4 2 2 3 3 4 4, , , , , , , , , , , ,x y x y x y x y x y x yb b b b b b t t t t t t= p
1 2 2 3 3 4 4 5, , , , , , ,
T
y x y x y x y yp p p p p p p p
MeanlinePrediction
Annulus Line & Pressure Ratio Optimization
Humboldt-Universität, Berlin, 25.5.2005
polyc, max ηp
6.0 max ≤iiΨ
1.1max ', ≤RiI
iM
8.0max , ≤SiI
iM
55.0max ≤Ri
iDF
55.0max ≤Si
iDF
58.0max ≥Ri
iDH
58.0max ≥Si
iDH 93.0 max , ≤ih
iC
285.0,
≤SNE s
M
M max Sp
s,N, i …1∈
1 0 2 0jy. p .≤ ≤
51 ,, j …∈
c max Πp
criteria
constraints
MeanlinePrediction
25SM %≥
ignored
Annulus Line & Pressure Ratio Optimization
Humboldt-Universität, Berlin, 25.5.2005
21.5886
22.0712
22.5538
23.0363
23.5189
24.0015
24.4841
24.9667
25.4492
25.9318
26.4144
26.897
Πi
*
MeanlinePrediction
E3E
human design
Annulus Line & Pressure Ratio Optimization
Polytropic Efficiency, ηηηηpoly
[%]
Su
rge
Ma
rgin
, S
M [
%]
Humboldt-Universität, Berlin, 25.5.2005
MeanlinePrediction
Annulus Line & Pressure Ratio Optimization
Humboldt-Universität, Berlin, 25.5.2005
Comparison: some numbers
18.2h839
(~8%)
10637
(~99%)10768
NLPQL 5CP
(Annulus)
94.1h
30.5h
19.3h
total
time
11909
(~28%)
42854
(~79%)54000
MIGA 5CP
(Annulus+PR)
5559
(~40%)
13743
(~86%)16000
MIGA 5CP
(Annulus)
4174
(~58%)
7166
(~90%)8000
MIGA 4CP
(Annulus)
feasibleconvergedfunction
evals
17min24
(~16%)
151
(100%)151
NLPQL 5CP
(Annulus)
best efficiency
MeanlinePrediction
Humboldt-Universität, Berlin, 25.5.2005
Off-Design
Process Overview
Throughflow
Humboldt-Universität, Berlin, 25.5.2005
-0.59293051
0.964143808
0.03 K
0.023 %
0.19 %
~2.1min
3.6 h
98
iSight
(Case1)
1.01857design parameter χ
0.83 KT criterion (∆T ≤ 1K)
-0.59design parameter ∆D
0.53 %η criterion (∆η/η ≤ 1%)
0.22 %Π criterion (∆Π/Π ≤ 1%)
~21mintime for one iteration
8 hoverall opt. time
No. iterations (feasible) 23
human
eng.
D,min
Tχ
η η
∆
∆ ∆Π Π ∆
multi criteriaproblem
nonlinearprogrammingproblem
Dmin u
χ∆ ,where ( )
2 22
100 100u Tη
η
∆ ∆Π = ⋅ + ⋅ + ∆
Π
best fastest
-0.5
1.0
1.2 K
0.544 %
0.063 %
~2.1min
1.6 h
46
iSight
(Case2)
0.95351
0.98 K
-0.62655
0.162 %
0.032 %
~2.1min
0.76 h
22
iSight
(Case3)
Throughflow
eng.
Humboldt-Universität, Berlin, 25.5.2005
20
20
20
20
20
20
20
20
20
20
40
40
40
40
40
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40 40
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200200
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200
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0A
dditio
nalD
evia
tion
[deg]
∆D
0.6 0.8 1.0 1.2 1.4
0.6 0.8 1.0 1.2 1.4
Pressure Loss Coefficient Factor [-]χ
0
20
40
60
80
100
120
140
160
180
200
ConvergedSolution
EngineeriSight
iSight
Throughflow
Humboldt-Universität, Berlin, 25.5.2005
Conclusions
• by nature, technical design problems are multi-criterion optimization problems
• multi-criterion optimization cuts down costs by releasing human design engineer from time-consuming parameter studies without taking him off the decision process
• multi-criterion optimization is a valuable tool for a variety of practical applications as
- optimal system design (active and passive)- identification- search for admissible solutions (stability)
• multi-criterion optimization finds better results than human designer
• integrated system design allows heterogeneous analysis tools on heterogeneous platforms